1. [Part 5] 1/53 Stochastic FrontierModels Heterogeneity Stochastic Frontier Models William Greene Stern School of Business New York University 0Introduction 1Efficiency Measurement 2Frontier Functions 3Stochastic Frontiers 4Production and Cost 5Heterogeneity 6Model Extensions 7Panel Data 8Applications

2. [Part 5] 2/53 Stochastic FrontierModels Heterogeneity Where to Next?  Heterogeneity: “Where do we put the z’s?” Other variables that affect production and inefficiency Enter production frontier, inefficiency distribution, elsewhere?  Heteroscedasticity Another form of heterogeneity Production “risk”  Bayesian and simulation estimators The stochastic frontier model with gamma inefficiency Bayesian treatments of the stochastic frontier model  Panel Data Heterogeneity vs. Inefficiency – can we distinguish Model forms: Is inefficiency persistent through time?  Applications

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4. [Part 5] 4/53 Stochastic FrontierModels Heterogeneity Swiss Railway Data

5. [Part 5] 5/53 Stochastic FrontierModels Heterogeneity Observable Heterogeneity  As opposed to unobservable heterogeneity  Observe: Y or C (outcome) and X or w (inputs or input prices)  Firm characteristics or environmental variables. Not production or cost, characterize the production process. Enter the production or cost function? Enter the inefficiency distribution? How?

6. [Part 5] 6/53 Stochastic FrontierModels Heterogeneity Shifting the Outcome Function Firm specific heterogeneity can also be incorporated into the inefficiency model as follows: This modifies the mean of the truncated normal distribution y i =  x i + v i - u i v i ~ N[0,  v 2 ] u i = |U i | where U i ~ N[  i,  u 2 ],  i =  0 +  1 z i,

7. [Part 5] 7/53 Stochastic FrontierModels Heterogeneity Heterogeneous Mean in Airline Cost Model

8. [Part 5] 8/53 Stochastic FrontierModels Heterogeneity Estimated Economic Efficiency

9. [Part 5] 9/53 Stochastic FrontierModels Heterogeneity How do the Zs affect inefficiency?

10. [Part 5] 10/53 Stochastic FrontierModels Heterogeneity Effect of Zs on Efficiency

11. [Part 5] 11/53 Stochastic FrontierModels Heterogeneity Swiss Railroads Cost Function

12. [Part 5] 12/53 Stochastic FrontierModels Heterogeneity One Step or Two Step 2 Step: 1. Fit Half or truncated normal model, 2. Compute JLMS u i, regress u i on z i Airline EXAMPLE: Fit model without POINTS, LOADFACTOR, STAGE 1 Step: Include z i in the model, compute u i including z i Airline example: Include 3 variables Methodological issue: Left out variables in two step approach.

13. [Part 5] 13/53 Stochastic FrontierModels Heterogeneity One vs. Two Step Efficiency computed without load factor, stage length and points served. Efficiency computed with load factor, stage length and points served. 0.8 0.9 1.0

14. [Part 5] 14/53 Stochastic FrontierModels Heterogeneity Application: WHO Data

15. [Part 5] 15/53 Stochastic FrontierModels Heterogeneity Unobservable Heterogeneity  Parameters vary across firms Random variation (heterogeneity, not Bayesian) Variation partially explained by observable indicators  Continuous variation – random parameter models: Considered with panel data models  Latent class – discrete parameter variation

16. [Part 5] 16/53 Stochastic FrontierModels Heterogeneity A Latent Class Model

17. [Part 5] 17/53 Stochastic FrontierModels Heterogeneity Latent Class Efficiency Studies  Battese and Coelli – growing in weather “regimes” for Indonesian rice farmers  Kumbhakar and Orea – cost structures for U.S. Banks  Greene (Health Economics, 2005) – revisits WHO Year 2000 World Health Report  Kumbhakar, Parmeter, Tsionas (JE, 2013) – U.S. Banks.

18. [Part 5] 18/53 Stochastic FrontierModels Heterogeneity Latent Class Application Estimates of Latent Class Model: Banking Data

19. [Part 5] 19/53 Stochastic FrontierModels Heterogeneity Inefficiency?  Not all agree with the presence (or identifiability) of “inefficiency” in market outcomes data.  Variation around the common production structure may all be nonsystematic and not controlled by management  Implication, no inefficiency: u = 0.

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21. [Part 5] 21/53 Stochastic FrontierModels Heterogeneity

22. [Part 5] 22/53 Stochastic FrontierModels Heterogeneity Nursing Home Costs  44 Swiss nursing homes, 13 years  Cost, Pk, Pl, output, two environmental variables  Estimate cost function  Estimate inefficiency

23. [Part 5] 23/53 Stochastic FrontierModels Heterogeneity Estimated Cost Efficiency

24. [Part 5] 24/53 Stochastic FrontierModels Heterogeneity A Two Class Model  Class 1: With Inefficiency logC = f(output, input prices, environment) +  v v +  u u  Class 2: Without Inefficiency logC = f(output, input prices, environment) +  v v  u = 0  Implement with a single zero restriction in a constrained (same cost function) two class model  Parameterization: λ =  u /  v = 0 in class 2.

25. [Part 5] 25/53 Stochastic FrontierModels Heterogeneity LogL= 464 with a common frontier model, 527 with two classes

26. [Part 5] 26/53 Stochastic FrontierModels Heterogeneity

27. [Part 5] 27/53 Stochastic FrontierModels Heterogeneity Heteroscedasticity in v and/or u y i =  ’x i + v i - u i Var[v i | h i ] =  v 2 g v (h i,  ) =  vi 2 g v (h i,0) = 1, g v (h i,  ) = [exp(  ’ h i )] 2 Var[U i | h i ] =  u 2 gu(hi,  )=  ui 2 g u (h i,0) = 1, g u (h i,  ) = [exp(  ’ h i )] 2

28. [Part 5] 28/53 Stochastic FrontierModels Heterogeneity Heteroscedasticity Affects Inefficiency

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30. [Part 5] 30/53 Stochastic FrontierModels Heterogeneity

31. [Part 5] 31/53 Stochastic FrontierModels Heterogeneity

32. [Part 5] 32/53 Stochastic FrontierModels Heterogeneity A “Scaling” Truncation Model

33. [Part 5] 33/53 Stochastic FrontierModels Heterogeneity Application: WHO Data

34. [Part 5] 34/53 Stochastic FrontierModels Heterogeneity Unobserved Endogenous Heterogeneity  Cost = C(p,y,Q), Q = quality Quality is unobserved Quality is endogenous – correlated with unobservables that influence cost  Econometric Response: There exists a proxy that is also endogenous Omit the variable? Include the proxy?  Question: Bias in estimated inefficiency (not interested in coefficients)

35. [Part 5] 35/53 Stochastic FrontierModels Heterogeneity Simulation Experiment  Mutter, et al. (AHRQ), 2011  Analysis of California nursing home data  Estimate model with a simulated data set  Compare biases in sample average inefficiency compared to the exogenous case  Endogeneity is quantified in terms of correlation of Q(i) with u(i)

36. [Part 5] 36/53 Stochastic FrontierModels Heterogeneity A Simulation Experiment Conclusion: Omitted variable problem does not make the bias worse.

37. [Part 5] 37/53 Stochastic FrontierModels Heterogeneity Sample Selection Modeling  Switching Models: y*|technology = b t ’x + v –u Firm chooses technology = 0 or 1 based on c’z+e e is correlated with v  Sample Selection Model: Choice of organic or inorganic Adoption of some technological innovation

38. [Part 5] 38/53 Stochastic FrontierModels Heterogeneity Early Applications  Heshmati A. (1997), “Estimating Panel Models with Selectivity Bias: An Application to Swedish Agriculture”, International Review of Economics and Business 44(4), 893-924.  Heshmati, Kumbhakar and Hjalmarsson Estimating Technical Efficiency, Productivity Growth and Selectivity Bias Using Rotating Panel Data: An Application to Swedish Agriculture  Sanzidur Rahman Manchester WP, 2002: Resource use efficiency with self-selectivity: an application of a switching regression framework to stochastic frontier models:

39. [Part 5] 39/53 Stochastic FrontierModels Heterogeneity Sample Selection in Stochastic Frontier Estimation Bradford et al. (ReStat, 2000):“... the patients in this sample were not randomly assigned to each treatment group. Statistically, this implies that the data are subject to sample selection bias. Therefore, we utilize a standard Heckman two-stage sample-selection process, creating an inverse Mill’s ratio from a first-stage probit estimator of the likelihood of CABG or PTCA. This correction variable is included in the frontier estimate....” Sipiläinen and Oude Lansink (2005) “Possible selection bias between organic and conventional production can be taken into account [by] applying Heckman’s (1979) two step procedure.”

40. [Part 5] 40/53 Stochastic FrontierModels Heterogeneity Two Step Selection Heckman’s method is for linear equations Does not carry over to any nonlinear model The formal estimation procedure based on maximum likelihood estimation – Terza (1998) – general results for exponential models with extensions to other nonlinear models – Greene (2006) – general template for nonlinear models – Greene (2010) – specific result for stochastic frontiers

41. [Part 5] 41/53 Stochastic FrontierModels Heterogeneity A Sample Selected SF Model d i = 1[  ′z i + w i > 0], w i ~ N[0,1 2 ] y i =  ′x i +  i,  i ~ N[0,   2 ] (y i,x i ) observed only when d i = 1.  i = v i - u i u i = |  u U i | =  u |U i | where U i ~ N[0,1 2 ] v i =  v V i where V i ~ N[0,1 2 ]. (w i,v i ) ~ N 2 [(0,1), (1,  v,  v 2 )]

42. [Part 5] 42/53 Stochastic FrontierModels Heterogeneity Alternative Approach Kumbhakar, Sipilainen, Tsionas (JPA, 2008)

43. [Part 5] 43/53 Stochastic FrontierModels Heterogeneity Sample Selected SF Model

44. [Part 5] 44/53 Stochastic FrontierModels Heterogeneity Simulated Log Likelihood for a Stochastic Frontier Model The simulation is over the inefficiency term.

45. [Part 5] 45/53 Stochastic FrontierModels Heterogeneity 2nd Step of the MSL Approach

46. [Part 5] 46/53 Stochastic FrontierModels Heterogeneity JLMS Estimator of u i

47. [Part 5] 47/53 Stochastic FrontierModels Heterogeneity WHO Efficiency Estimates OECD Everyone Else

48. [Part 5] 48/53 Stochastic FrontierModels Heterogeneity

49. [Part 5] 49/53 Stochastic FrontierModels Heterogeneity WHO Estimates vs. SF Model

50. [Part 5] 50/53 Stochastic FrontierModels Heterogeneity

51. [Part 5] 51/53 Stochastic FrontierModels Heterogeneity Sample Selection in a Stochastic Frontier Model TECHNICAL EFFICIENCY ANALYSIS CORRECTING FOR BIASES FROM OBSERVED AND UNOBSERVED VARIABLES: AN APPLICATION TO A NATURAL RESOURCE MANAGEMENT PROJECT Boris Bravo-Ureta University of Connecticut Daniel Solis University of Miami William Greene Stern School of Business New York University

52. [Part 5] 52/53 Stochastic FrontierModels Heterogeneity  Component II - Module 3 focused on promoting investments in sustainable production systems with a budget of US $7.6 million (Bravo-Ureta, 2009).  The major activities undertaken with beneficiaries: training in business management and sustainable farming practices; and the provision of funds to co- finance investment activities through local rural savings associations (cajas rurales). Component II - Module 3

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